Let $X/k$ be a scheme over base field $k$ and $G$ an group scheme (also over $k$) acting on $X$ (ie there is an algebraic map $\sigma: G \times X \to X$ satisfying some compatibility conditions).
Let $x \in X(k)$ be a rational point of $X$. Then the orbit $\mathcal{O}_x$ of $x$ under $G$ is given as scheme theoretic image under composition map $ G \times_k x \xrightarrow{\text{id}_G \times x} G \times X\xrightarrow{\sigma} X$
(...here we interpret $x$ as canonical map $x:\operatorname{Spec } \kappa(x) \to X$)
Remark: Note that it seems that to define an orbit of a "point" of $X$ there is no reason to assume $x$ to be a rational point (ie $\kappa(x)=k$) or even to be a closed point, ie such that the residue field $\kappa(x)$ is finite extension of $k$. It seems that we can declare for every $k$-scheme $S$ which admits a $k$-map $s:S \to X$ (...so is a $S$-valued point of $X$ in sense of scheme as point functor philosophy) then we can declare the orbit of $s$ -denoted by $\mathcal{O}_s$ - to be as before the scheme theoretic image of the composition map $ G \times_k S \xrightarrow{\text{id}_G \times s} G \times X\xrightarrow{\sigma} X$.
Now in Hoskins' notes there are given two classical key results (see pages 18 & following) on action by algebraic groups on schemes telling following:
Proposition 3.15. Let $G$ be an affine algebraic group acting on scheme $X$. The orbits of closed points are locally closed subsets of $X$, hence can be identified with the corresponding reduced locally closed subschemes.
and the "dimension count formula":
Proposition 3.20. Let $G$ be an affine algebraic group acting on a scheme $X$. For rational point $x \in X(k)$, we have $$\dim(\mathcal{O}_x) = \dim(G)-\dim(\text{Stab}_G(x)) $$
Question: To which extend these two results are still true if we would consider $G$ to be group scheme (over base field $k$) and instead to consider only rational points $x \in X(k)$, to take any $S$-valued point in above sense interpreting $X$ as point functor, ie $k$-morphisms $s: S \to X$ for $S$ a $k$-scheme?
To avoid pathologies let add finiteness conditions, so say $S$ is Noetherian, or say even of finite type over $k$.
Considerations: Going through the proofs, Proposition 3.15 used Chevalley's theorem on constructibility of image (holds for $f: X \to Y$ morphism of finite type of noetherian schemes; so if $G$ & $S$ are of finite type over $k$ we are on safe side anyway.
Prop. 3.20 used a fiber dimension theorem assuring that that there exist a nontrivial open locus $U \subset \mathcal{O}_s$ over which orbit map $\sigma \circ (\text{id}_G \times s$ is flat once assuming $\mathcal{O}_s$ to be reduced; but that's ok for dimension count. Then by transitivity of action we can flatness "spread" everywhere.
The critical part is do we really unavoidably need $G \times_k S$ and $\mathcal{O}_s$ to be of finite type over $k$? (ie to be really "variety like). Hartshorne's fiber counting theorem (Ch. III, 9.5) requires it, but I'm not sure if that's exactly the weakest form of such type of a "fiber counting theorem". Are there maybe weaker forms allowing to generalize the result.
If not, it seems the only potential obstruction of generalize these results is to assume all involved objects to be of finite type over $k$. For $S$-point $s:S \to X$ that's ok to assume it to be of finiet type over $k$, but to assure this for scheme theoretic image $\mathcal{O}_s$ looks to be not quite easy, since the quasicoherent ideal defining it can be rather complicated, see again here.
What also surprises me a bit is that it seems that we not need to worry on if $G$ is a non smooth group scheme (=non reduced for group schemes). For instance, in dimension formula count (purely about topology) we can seemingly just pass to reduced schemes and so circumvent the problem of non smooth group scheme $G$.
